Abstract
In this paper we consider the two-dimensional Dirichlet and impedance boundary value problems for the Helmholtz equation, Δu + k 2 u = 0, in a non-locally perturbed half-plane with a periodic Lipschitz boundary. The Dirichlet problem arises in a study of time-harmonic acoustic scattering of an incident field by a sound-soft, non-smooth (Lipschitz) periodic surface where the total field u t (the sum of the incident field u i and the scattered field u) vanishes. The impedance problem, with the boundary condition ∂u/∂v+iλu = 0, where λ ∈ ℂ is a constant, models acoustic or electromagnetic scattering (in both polarization cases) by a one-dimensional Lipschitz periodic boundary of finite surface impedance.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
T. Arens, S.N. Chandler-Wilde, and K.O. Haseloh, Solvability and spectral properties of integral equations on the real line. II. L p spaces and applications, J. Integral Equations Appl. 15 (2003), 1–35.
G. Bao, D.C. Dobson, and J.A. Cox, Mathematical studies in rigorous grating theory, J. Opt. Soc. Amer. A12 (1995), 1029–1042.
S.N. Chandler-Wilde, C.R. Ross, and B. Zhang, Scattering by infinite one-dimensional rough surfaces, Proc. Roy. Soc. London A 455 (1999), 3767–3787.
X. Chen and A. Friedman, Maxwell’s equations in a periodic structure, Trans. Amer. Math. Soc. 323 (1991), 465–507.
D. Dobson and A. Friedman, The time-harmonic Maxwell equations in a doubly periodic structure, J. Math. Anal. Appl. 166 (1992), 507–528.
J. Elschner and M. Yamamoto, An inverse problem in periodic diffractive optics: reconstruction of Lipschitz grating profiles, preprint no. 718, Weierstrass Institute für Angewandte Analysis und Stochastik, Berlin, 2002.
A. Kirsch, Diffraction by periodic structures, in Inverse Problems in Mathematical Physics, L. Paivarinta and E. Somersalo (eds.), Springer-Verlag, 1993, 87–102.
C.R. Ross, Direct and Inverse Scattering by Rough Surfaces, PhD thesis, Brunei University, 1996.
R.H. Torres and G.V. Welland, The Helmholtz equation and transmission problems with Lipschitz interfaces, Indiana Univ. Math. J. 42 (1993), 1457–1485.
G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equations in Lipschitz domains, J. Functional Anal. 59 (1984), 572–611.
B. Zhang, G. Yan, and C. Miao, Diffraction by Lipschitz periodic structures (in preparation).
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer Science+Business Media New York
About this chapter
Cite this chapter
Zhang, B., Yan, G. (2004). Integral Equation Methods for Scattering by Periodic Lipschitz Surfaces. In: Constanda, C., Largillier, A., Ahues, M. (eds) Integral Methods in Science and Engineering. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8184-5_42
Download citation
DOI: https://doi.org/10.1007/978-0-8176-8184-5_42
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6479-8
Online ISBN: 978-0-8176-8184-5
eBook Packages: Springer Book Archive